Ball indenter utilizing fea solutions for property evaluation

ABSTRACT

An automated indentation system based on finite element solutions or performing a non-destructive compression test by loading a compressive indentation load (P) and calculating a elastic modulus (E) and a yield strength (σ o ), and a hardening exponent (n) from measured indentation depth (h t ) and indentation load (P), and unloading slope (S). The system comprises a stepmotor control system ( 1 ), a measurement instrumentation ( 2 ) having a load cell ( 15 ), laser displacement sensor ( 17 ) for measuring the indentation depth and ball indenter ( 18 ), a data acquisition system ( 3 ) having an signal amplifier for amplifying and filtering signals from the load cell ( 15 ) and laser displacement sensor ( 17 ), and a control box ( 4 ) pre-stored computer programming algorisms for adjusting and controlling the moving speed and direction of stepmotor ( 12 ). The control box ( 4 ) enables storing and retrieving the signals of measured data and material properties, and plotting the graphs of load-depth curve and stress-strain curves based on the signal data. The procedure of computer programming algorithm is as follows: First, the Young&#39;s modulus E is computed from Eq. (29) by using slope S and initially guessed values of n and ε o . Then, c 2 , εp and σ are calculated as many as the number of load and depth data. From these, the values of n, K, σ o  and ε o  are calculated from stress-strain relation. And then updated E, d, c 2 , ε p , σ, n, K, σ o  and ε o  are repeatedly calculated until the updated ε o  and n are converged within the tolerance.

BACKGROUND OF THE INVENTION

[0001] 1. Field of the Invention

[0002] The present invention relates to a ball indentation tester andtesting technique being used to measure the material properties whentensile test cannot be applied; welding parts with continuous propertyvariation, brittle materials with unstable crack growth duringpreparation and test of specimen, and the parts in present structuraluse. More particularly, indentation test is non-destructive and easilyapplicable to obtain material properties. A new numerical indentationtechnique is invented by examining the finite element solutions based onthe incremental plasticity theory with large geometry change. Theload-depth curve from indentation test successfully converts to astress-strain curve.

[0003] 2. Background of the Related Art

[0004] While indentation test is non-destructive and easily applicableto obtain material properties, the test result is difficult to analyzebecause of complicated triaxial stress state under ball indenter. Forthis reason, the indentation test is inappropriate to measure variousmaterial properties. Thus, it is used to obtain merely hardness.Recently, however, this kind of difficulty is greatly overcome both byfinite element analyses of subindenter stress and deformation fields,and by continuous measurement of load and depth. As a result,stress-strain relation can be obtained from analysis of load-depthcurve.

[0005] An automated indentation test gives a stress-strain curve frommeasured load-depth data. FIG. 1 shows a schematic profile ofindentation. Here h_(t) and d_(t) are ideal indentation depth andprojected diameter at loaded state, and h_(p) and d_(p) are plasticindentation depth and projected diameter at unloaded state. With anindenter of diameter D, the following relation is delivered fromspherical geometric configuration.

d _(t)=2 {square root}{square root over (h_(t)D−h_(t) ²)}  (1)

[0006] Assuming that “projected” indentation diameter at loaded andunloaded states remains the same as shown in FIG. 2, Hertz expressed d(=d_(t)=d_(p) under this assumption) as follows. $\begin{matrix}{d = {2.22\left\{ {\frac{P}{2}\frac{r_{1}r_{2}}{r_{2} - r_{1}}\left( {\frac{1}{E_{1}} - \frac{1}{E_{2}}} \right)} \right\}^{1/3}}} & (2)\end{matrix}$

[0007] where r₁ and r₂ are indentation radius of indenter and specimenat unloaded state, and E₁ and E₂ are Young's modulus of indenter andspecimen, respectively. If the indenter is rigid, r₁=D/2 and r₂ is afunction of d and h_(p).

[0008] Substituting these into Eq. (3) gives: $\begin{matrix}{d = \left\lbrack \frac{0.5C\quad D\left\{ {h_{p}^{2} + \left( {d/2} \right)^{2}} \right\}}{h_{p}^{2} + \left( {d/2} \right)^{2} - {h_{p}D}} \right\rbrack^{1/3}} & (3)\end{matrix}$

[0009] where C is 5.47P (E₁ ⁻¹+E₂ ⁻¹).

[0010] Tabor brought the experimental conclusion that equivalent(plastic) strain “at the (Brinell and Micro Vickers) indenter contactedge” is given by: $\begin{matrix}{ɛ_{p} = {0.2\left( \frac{d}{D} \right)}} & (4)\end{matrix}$

[0011] where d is calculated from Eq. (2). But, Haggag et al. ignoredpile-up and sink-in of material. They simply calculated the indentationdiameter d with Eq. (1) at loaded state and plastic diameter d_(p) withEq. (3) at unloaded state, and plastic strain with Eq. (4) bysubstituting d_(p) for d.

[0012] Mean contact pressure p_(m) is defined by p_(m)=4P(πd²), where Pis the compressive indentation load. Then constraint factor ψ, which isa function of equivalent plastic strain, is defined as the ratio betweenmean contact pressure and equivalent stress.

ψ(ε_(o))≡p _(m)/σ  (5)

[0013] Hence, the equivalent stress is expressed in the form:$\begin{matrix}{\sigma = \frac{4P}{\pi \quad d^{2}\psi}} & (6)\end{matrix}$

[0014] Note that, in a strict sense, both equivalent plastic strain andequivalent stress are functions of location within the subindenterdeformed region as well as deformation intensity itself. Thus constraintfactor ψ is also a function of location. Francis classified theindentation states into three regions and presented the empiricalformula for ψ with indentation test results for the various materialstaken into consideration.

[0015] (1) Elastic region with recoverable deformation

[0016] (2) Transient region with elastic-plastic deformation

[0017] (3) Fully plastic region with dominant plastic deformation

[0018] Haggag et al. calculated stress-using d_(p) instead of d in Eq.(6), and they modified Francis' constraint factor considering thatconstraint factor is a function of strain rate and strain hardening.$\begin{matrix}{\psi = \left\{ \begin{matrix}{\quad 1.12} & {\quad {\varphi \leq 1}} \\{\quad {1.12 + {\tau ln\varphi}}} & {\quad {1 \leq \varphi \leq 27}} \\{\quad \psi_{\max}} & {\quad {\varphi \geq 27}}\end{matrix} \right.} & \text{(7a)}\end{matrix}$

ψ_(max)=2.87α_(m)   (7b)

τ≡(ψ_(max)−1.12)/ln27   (7c)

[0019] where α_(m) is constraint factor index. It is proportional tostrain rate, and has the value of 1 for the material with low strainrate. By investigating the experimental results, Francis suggested anormalized variable φ in the form: $\begin{matrix}{\varphi = \frac{ɛ_{p}E_{2}}{0.43\sigma}} & (8)\end{matrix}$

[0020] Since equivalent strain in Eq. (4) is the value at the indentercontact edge, all the values in Eqs. (5)-(8) implicitly mean values alsoat the indenter contact edge.

[0021] For spherical indenter, the following relation called Meyer's lawholds between applied load P and indentation projected diameter d.

P=kd^(m)   (9)

[0022] where k and m are material constants when indenter diameter D isfixed, and m is Meyer's index generally in the range of 2 to 2.5.

[0023] Meyer's experiment reveled that index m is independent ofdiameter D, and k decreases with increasing D.

A=k₁D₁ ^(m−2)=k₂D₂ ^(m−2)k₃D₃ ^(m−2)=  (10)

[0024] where A is a constant. Substituting this into Eq. (9) gives:$\begin{matrix}{\frac{P}{d^{2}} = {A\left( \frac{d}{D} \right)}^{m - 2}} & (11)\end{matrix}$

[0025] Equation (6) converts to Eq. (12) by Eq. (11). $\begin{matrix}{\sigma = {\frac{4A}{\pi\psi}\left( \frac{d}{D} \right)^{m - 2}}} & (12)\end{matrix}$

[0026] After replacing d with d_(t) in Eq. (11), Haggag et al.calculated yield strength σ_(o) from the following relation of yieldstrength and slope A that George et al. obtained from experiment.

σ_(o)=β_(m)A   (13)

[0027] where β_(m) is a material constant. The value of β_(m) in steelis about 0.229, which comes from analysis of tensile yield strength andA.

[0028] Rice and Rosengren proposed a stress-strain relation in piecewisepower law form. $\begin{matrix}{\frac{ɛ_{t}}{ɛ_{o}} = \left\{ \begin{matrix}{\quad \frac{\sigma}{\sigma_{o}}} & {\quad {{{fo}\quad r\quad \sigma} \leq \sigma_{o}}\quad} \\{\quad \left( \frac{\sigma}{\sigma_{o}} \right)^{n}} & {\quad {{{{fo}\quad r\quad \sigma} > \sigma_{o}};{1 < n \leq \infty}}}\end{matrix} \right.} & (14)\end{matrix}$

[0029] where σ_(o) is yield strength, ε_(o)=σ_(o)/E yield strain and nstrain hardening exponent. Total strain ε_(t) is decomposed into elasticand plastic strains (ε_(t)=ε_(e)+ε_(p)).

[0030]FIG. 3 shows the calculation process of the material properties byHaggag's indentation method. In the approach of Haggag et al., eachrepetition of loading and unloading provides one point of stress-straindata points. Thus a single indentation test usually picks up total only6-7 data point. The approach also requires prior material constants fromextra tensile tests.

[0031] The Haggag's model for the SSM system adopts the indentationtheories of Francis and Tabor established on the experimentalobservations and some analyses. Haggag's approach requires priormaterial constants from extra tensile tests, which is one of theshortcomings.

[0032] The SSM system gives stress-strain curves through regression ofload-depth data obtained from 6-7 times repetitive loading andunloading. This insufficient number of data often leads to inaccurateregression. Above all, the most critical issue in Haggag's approach isthat subindenter stress field from deformation theory is far from thereal one.

SUMMARY OF THE INVENTION

[0033] In order to overcome the disadvantages as described above, thepresent invention provides an automated indentation system forperforming a compression test by loading a compressive indentation load(P). Then, an elastic modulus (E), a yield strength ((σ_(o)), and ahardening exponent (n) are calculated based on measured indentationdepth (h_(t)) and indentation load (P), and unloading slope (S).

[0034] The automated indentation system of the present inventioncomprises a stepmotor control system (1), measurement instrumentation(2) data acquisition system (3) and control box (4).

[0035] The stepmotor (12) is adopted for precisely controlling atraveling distance and minimizing vibration of motor.

[0036] The measurement instrumentation (2) consists of a load cell (15),a laser displacement sensor (17) for measuring the indentation depth,and a ball indenter (18).

[0037] The data acquisition system (3) includes a signal amplifier foramplifying and filtering signals received from the load cell (15) andlaser displacement sensor (17).

[0038] The control box (4) is pre-stored computer programming algorismsfor adjusting and controlling moving speed and direction of thestepmotor (12). It also enables to perform calculations and plot thegraphs of load-depth curve or strain-stress curves according to theamplified signal data, and store and retrieve the measured signal data,material properties and produced data.

[0039] The stepmotor control system (1) comprises a cylindrical linearactuator having a ball screw (14) and backlash nut (16) for suppressingbacklash, a flexible coupling (13) being connected to the ball screw(14) and stepmotor (12) for constraining the rotation and highrepeatability. The stepmotor control system (1) also enables to controlacceleration/deceleration of the stepmotor (12) and regulating velocitywith repeatability of 3˜5%.

[0040] The load cell (15) is specified based on the performance offinite element simulation of indentation test. The indentation load (P)is dependent on the ball size and material properties, and maximumindentation load is under 100 kgf for 1 mm indenter.

[0041] The laser displacement sensor for measuring indentation depth isconnected parallel to a linear actuator, and measurement range of laserdisplacement sensor (17) is 4 mm and resolution is 0.5 μm.

[0042] The ball indenter (18) is an integrated spherical indenter beingmade of tungsten carbide (WC) for precisely measuring an indented depth,and a diameter of indenter tip is 1 mm.

[0043] Generally, the measured indentation depth (h_(exp)) contains anadditional displacement due to system compliance (h_(add)). Therefore,in order to obtain an accurate indentation depth, the practicalindentation depth is compensated by a displacement relationship betweenthe measured indentation depth (h_(exp)) and an actual indentation depth(h_(FEM)) obtained from FEA.

[0044] A computer programming algorism is provided for performing anautomated indentation test by loading a compressive indentation load(P), calculating an elastic modulus (E) and a yield strength (σ_(o)),and a hardening exponent (n) from measured indentation depth (h_(t)) andindentation load (P), and unloading slope (S), then plotting astress-strain curve of the indented material.

[0045] The process of computer programming algorism comprises the stepsof: inputting data of measured indentation depth (h_(t)), load (P) andunloading slope (S) from pre-stored data, computing a Young's modulus(E) from unload slope and initially guessed values of n and ε_(o),computing indentation diameters (d) from c² equation as many as thenumber of load and depth data, computing equivalent plastic strains(ε_(p)) and equivalent stresses (σ) according to the calculatedindentation diameters (d), computing values of strain hardening exponent(n) and K from stress-strain relation, computing a yield stress (σ_(o))and strain (ε_(o)) computing updated E, d, c², ε_(p), σ, n, K, σ_(o),and ε_(o) until the updated ε_(o) and n are converged within thetolerance, and outputting material properties (E, σ_(o), n) and plottingthe stress-strain curve.

BRIEF DESCRIPTION OF THE DRAWINGS

[0046]FIG. 1 illustrates a schematic profile of typical indentation.

[0047]FIG. 2 is a projected indentation diameter at loaded and unloadedstates.

[0048]FIG. 3 is a calculation process of the material properties.

[0049]FIG. 4 is a finite element (FE) model for a ball indentation test.

[0050]FIG. 5 is the distribution of equivalent plastic strain ε_(p)along radial direction r for strain hardening exponent n=10, frictioncoefficient f=0.0, 0.1 and 0.2, d/D=0.5 at specific depth l/D.

[0051]FIG. 6 illustrates a plastic strain curves at the new dataacquisition point for various strain hardening exponents with theprojected contact diameter.

[0052]FIG. 7 is a regression curves of c² against an indentation depthfor various values of n with f=0.1.

[0053]FIG. 8 is a FE solution and corresponding regression curves ofequivalent plastic strain ε_(p) against the projected contact diameterfor various values of n at l/D=1% and 2 r/d=0.4 for f=0.1.

[0054]FIG. 9 is a graph for the constraint factor ψ against projectedcontact diameter curve for various values of n.

[0055]FIG. 10 is the generated curves with a constraint factor ψ againstthe projected contact diameter as solid lines

[0056]FIG. 11 is a distribution of equivalent plastic strain along theradial direction for three values of yield strength at d/D=0.5 withYoung's modulus and fixed strain hardening exponent.

[0057]FIG. 12 is the relationships between c² and h_(t)/D for variousvalue of yield strain.

[0058]FIG. 13 is the effect of yield strain on ε_(o) vs. d/D curves.

[0059]FIG. 14(a) and 14(b) are the effect of yield strain on constraintfactor ψ vs. d/D curves, for the transition and fully plastic region,respectively.

[0060]FIG. 15(a) is the load-depth curves obtained by FEA for a varietyof material hardening exponent n with given Young's modulus and yieldstrain.

[0061]FIG. 15(b) is corresponding unloading slopes, S, abscissar=P/P_(max) measuring the portion of unloading curve used for linearregression.

[0062]FIG. 16(a) is the variation of k₁ with respect to n for variousvalues of E.

[0063]FIG. 16(b) is the extrapolation of k₁ with respect to n forvarious values of E.

[0064]FIG. 17(a) is the FEA relationships between E and S/d for variousvalues of n, when indentation depth is given as h_(t)/D=0.06.

[0065]FIG. 17(b) illustrates linear variation of b with 1/n, when a isfixed as 6200.

[0066]FIG. 18 is presently invented computing process of the materialproperties.

[0067]FIGS. 19 and 20 are the strain-stress curves via computing processof the present invention

[0068]FIG. 21 is an automated indentation system with a stepmotorcontrol system, measurement instrumentation, data acquisition system andcontrol box of the present invention.

[0069]FIG. 22 is a front view of automated indentation system of anembodiment of the present invention.

[0070]FIG. 23 illustrates integrated ball indenters

[0071]FIG. 24 illustrates a controlling box and a display of measuredload-depth curve

[0072]FIG. 25 illustrates the experimentally measured P-h curves aftercorrecting h_(add).

DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENT

[0073] The objectives and advantages of the present invention will bemore clearly understood through the following detailed descriptionsaccompanying with the drawings.

[0074] As shown in FIG. 4, a finite element (FE) model is provided for aball indentation test. The FE analysis of large deformation is performedby using isotropic elasto-plastic material, which follows the J₂ flowtheory. Considering both loading and geometric symmetry, the four nodesof axisymmetric elements CAX4 (ABAQUS, 2002) are applied. Thepreliminary analyses are revealed that the eight nodes CAX8 element hasa trouble of discontinuous equivalent plastic strain at its mid-node.The lower degree of CAX4 shape function is supplemented by placing fineelements with size 0.25% of indenter diameter at the material contactsurface. MPC (Multi-Point Constraints) is conveniently used at thetransition region where element size changes. But constrained mid-nodesof MPC tend to give discrete stress and strain values. Thus, thetrapezoidal elements are adopted in the transition region near contactsurface, and use MPC in the transition region far from the contactsurface. The FE model of specimen and indenter consist of about 2300 and630 elements, respectively. The contact surfaces are also placed at bothmaterial and indenter surfaces. Axisymmetric boundary conditions areimposed on the nodes on axisymmetric axis. The indenter moves down topenetrate the material with the bottom fixed. The diameter of indenteris 1 mm, Young's modulus 650 GPa.

[0075] Indentation theories of Matthews and Hill et al. are based on thedeformation theory of plasticity. Although the indentation theory iseasier to develop with deformation than with incremental plasticitytheory, the two theories produce quite different subindenterdeformations. For example, the maximum stress occurs at the bottom partof indentation center for deformation theory, while it occurs at thesurface part 0.4 d away from indentation center for incremental theory.Here d is the projected contact diameter, which takes the effect ofpile-up and sink-in into consideration.

[0076]FIG. 5 shows distribution of equivalent plastic strain ε_(p) alongradial direction r for strain hardening exponent n=10, frictioncoefficient f=0.0, 0.1 and 0.2, d/D =0.5 at specific depth l/D. Here ris the projected distance from indentation center, l is the distancefrom material surface to observation depth, and d/D is the ratio ofprojected contact diameter to indenter diameter. Equivalent plasticstrain oscillates at surface due to contact problem. Oscillation and theeffect of friction coefficient decreases as observing position movesdownward from the surface. Thus, a new probing depth is proposed to bebeneath 1% of indenter diameter from the surface (l/D=0.01). This depthis still near the surface, yet with negligible contact problem.

[0077] Distribution of equivalent strain is affected by frictioncoefficient as shown in FIG. 5. The value of friction coefficientbetween metals is commonly 0.1 to 0.2, but difficult to measure exactly.This is because it depends on the environmental factors such astemperature and humidity. Tabor suggested the acquisition point ofequivalent plastic strain as r/(d/2)=1; i.e. contact edge.

[0078] However it is difficult to extract accurate stress-strainrelationship from the suggested point because of sharp strain gradientand tangible effect of friction coefficient. Hence, a selection is madefor the data acquisition point at 0.4 d apart from indentation center.This new point features i) negligible effect of friction coefficient,ii) quite gentle strain gradient and iii) extended strain range by afactor of five. The ground of this arbitrary selection is that theequivalent stress and strain at any point should be on uniaxialstress-strain curve, even though the equivalent stress and strain ofmaterial vary point to point, and also with indentation depth.Deformation and incremental plasticity theories provide thoroughlydifferent stress and strain values at this new point. Consequently a newset of indentation governing equations should be developed using flowtheory.

[0079]FIG. 6 shows the plastic strain curves at the new data acquisitionpoint for various strain hardening exponents with the projected contactdiameter. Here friction coefficient f is 0.1, yield strength σ_(o) 400MPa, Young's modulus E 200 GPa. The plastic strain curve starts onlyslightly off the origin, which indicates that plastic deformation takesplace even at shallow indentation. The plastic strain increases withstrain hardening exponent, since material with larger n value deformsmore easily. The data acquisition point in the prior indentation theorywas the contact edge where maximum plastic strain can reach 0.2 underfull indentation of hemispheric ball. But as full indentation isimpractical, the maximum strain actually achieved is rather smaller thanthis maximum. Compared with the prior theory, the present approachprovides the strain value five times greater at the same indentationdepth; for example, exceeding 0.5 at d/D=0.5.

[0080] A new indentation theory is presented on the basis of FEsolutions from the new optimal data acquisition point, 2r/d=0.4 andl/D=1%, where the frictional effect is ignored and obtained maximumstrain for a given indentation depth. The actual projected contactdiameter with pile-up and sink-in in consideration is calculated fromthe geometric shape of a sphere.

d=2{square root}{square root over (hD−h²)}==2{square root}{square rootover (c²h_(t)D−(c²h_(t))²)}  (15)

[0081] Here h is actual indentation depth due to pile-up and sink-in,h_(t) is nominal depth measured from the reference surface (=originalmaterial surface), and c² is defined as: $\begin{matrix}{c^{2} \equiv \frac{h}{h_{t}}} & (16)\end{matrix}$

[0082] From FIG. 6, equivalent plastic strain can be approximated in thefunctional form below when the values of yield strength and Young'smodulus are fixed. $\begin{matrix}{ɛ_{e\quad q}^{p\quad l} = {{f_{o}^{ɛ}(n)}\left( \frac{d}{D} \right)^{f_{i}^{ɛ}{(n)}}}} & (17)\end{matrix}$

[0083] As mentioned above, Haggag's indentation theory necessitatesprior material constants, for finding yield strength, from extra tensiletests. This complicates evaluation of material properties by usingindentation test. An attempt is made to eliminate the extra process thruinterrelationship of material properties. Piecewise power law relation(14) for plastic deformation becomes: $\begin{matrix}{\sigma = {{\sigma_{o}\left( \frac{ɛ_{t}}{ɛ_{o}} \right)}^{1/n} = {K\quad ɛ_{t}^{1/n}}}} & (18)\end{matrix}$

[0084] where K is obtained by the regression of stress-strain data. AsEq. (18) is also valid for σ=σ_(o),

σ_(o)=Kε_(o) ^(1/n)   (19)

[0085] Elastic stress-strain relation at the moment of yielding is Eq.(20).

σ_(o)=Eε_(o)   (20)

[0086] Substituting (20) into Eq. (19) produces $\begin{matrix}{\sigma_{o} = {\left( \frac{K^{n}}{E} \right)^{\frac{1}{n - 1}} = {E\left( \frac{K}{E} \right)}^{\frac{n}{n - 1}}}} & (21)\end{matrix}$

[0087] Consequently, the n and K are calculated from the regression ofthe stress-strain relation of Eq. (18) if the stress corresponding tostrain is accurately predicted, and σ_(o) is obtained from Eq. (21).

[0088] First, the FE analyses are performed for various values of strainhardening exponent with yield strain fixed to verify above formulae. Itis difficult to measure the actual projected indentation diameter d.Thus, an approach is chosen for calculating d from Eq. (15) with c²obtained from regression of FE solutions.

[0089]FIG. 7 illustrates regression curves of c²against indentationdepth for various values of n with f=0.1. Herein, c² is calculated fromthe ratio of actual indentation depth to nominal indentation depth byits definition in Eq. (16). FIG. 7 reveals c² is a function ofindentation depth unlike the outcome of Matthews and Hill et al. inwhich c² was a constant for a given value of n. At the outset ofindentation, c² starts from the theoretical value 0.5 since elasticdeformation is initially dominant. Then c² increases with indentationdepth as plastic deformation becomes dominant. As pile-up occurs moreeasily with higher value of n, c² increases with n for a givenindentation depth, which is consistent with the results of Matthews,Hill and Norbury and Samuel. Note that even in sufficiently indentedfully plastic state, c² keeps slightly increasing, instead of saturatingto a constant value, with indentation depth. Overall the priorindentation theories should be revised such that c² is a function ofindentation depth as well as strain hardening exponent n. The FEsolutions of c² in FIG. 7 can be expressed with the following equation.

c ² =f _(o) ^(c)(n)+f ₁ ^(c)(n)1n(100 h_(t) /D)   (22)

f _(o) ^(c)(n)=a _(ot) ^(c) n ^(−i) ;a _(ot) ^(c)=(1.09, −1.21, 1.13)

f ₁ ^(c)(n)=a _(1t) ^(c) n ^(−i) ;a _(1t) ^(c)=(0.104, −0.323, 0.405)

[0090] where a is a coefficient of polynomial function.

[0091]FIG. 8 shows the FE solutions and corresponding regression curvesof equivalent plastic strain against the projected contact diameter forvarious values of n at l/D=1% and 2 r/d=0.4 for f=0.1. Equation (23) isthe regression formula of ε_(p) expressed as a function of d and n. Thecurve from Eq. (23) for n=8.5 is also compared with the FE solution inFIG. 8. The good agreement indicates that Eq. (23) successfullycharacterizes ε_(p) as a function of d and n. $\begin{matrix}{{ɛ_{p} = {{f_{o}^{ɛ}(n)}\left( \frac{d}{D} \right)^{f_{i}^{ɛ}{(n)}}}}{{{f_{o}^{ɛ}(n)} = {a_{o\quad i}^{ɛ}n^{- i}}};{a_{o\quad i}^{ɛ} = \left( {1.82,{- 5.82},6.92} \right)}}{{{f_{1}^{ɛ}(n)} = {a_{1\quad i}^{ɛ}n^{- i}}};{a_{1\quad i}^{ɛ} = \left( {1.45,{- 0.641},{- 0.233}} \right)}}} & (23)\end{matrix}$

[0092]FIG. 9 is the constraint factor ψ against projected contactdiameter curve for various values of n. At the outset of indentation, ψincreases nonlinearly with d/D. Then, as plastic deformation becomesdominant after substantial indentation, it shows a linear relation withd/D (FIG. 10). Furthermore ψ increases with n, which is consistent withthe results of Matthews and Tirupataiah. But more importantly, even insufficiently indented fully plastic state, ψ keeps slightly decreasing,instead of saturating to a constant value, with indentation depth. Inthe range of d/D≧0/15, ψ can be given by the following linearexpression. $\begin{matrix}{{{\psi = {{f_{o}^{\psi}(n)} + {{f_{1}^{\psi}(n)}\left( \frac{d}{D} \right)}}},\left( {{d/D} \geq 0.15} \right)}{{{f_{o}^{\psi}(n)} = {a_{o\quad i}^{\psi}n^{- i}}};{a_{o\quad i}^{\psi} = \left( {3.06,{- 4.4},4.19} \right)}}{{{f_{1}^{\psi}(n)} = {a_{1\quad i}^{\psi}n^{- i}}};{a_{1\quad i}^{\psi} = \left( {{- 0.227},0.317,1.25} \right)}}} & (24)\end{matrix}$

[0093] The generated curves with the above expression are presented inFIG. 10 as solid lines. Equivalent stress (at the new data acquisitionpoint) can be obtained by substituting ψ of Eq. (24) into Eq. (6).

[0094]FIG. 11 shows the distribution of equivalent plastic strain alongthe radial direction for three values of yield strength (σ_(o)=200, 400,800 MPa) at d/D=0.5 with Young's modulus (E=200 GPa) and strainhardening exponent (n=10) fixed. In such cases, the value of ε_(p) isobserved to decrease with increasing yield strength for a givenindentation depth. To clarify this kind of tendency, Young's modulus isadditionally varied below.

[0095]FIG. 12 shows the relationships between the c² and h_(t)/D forvarious value of yield strain. It is noteworthy that identical yieldstrain produces the same curve regardless of absolute values of σ_(o)and E; that is only the ratio σ_(o)/E=ε_(o) matters. The c² curve movesdownward with increasing yield strain since the higher yield strainprolongs the initial dominance of elastic deformation. Yield straindetermines the position of curve but barely affect the form of curveafter transition region. For given n=10, each curves are almost parallelin fully plastic region.

[0096] Hence, it may be selected ε_(o) and n as two separate variablesfor governing the deformation characteristics of indentation intransition and fully plastic region, respectively. With separation ofvariable approach, c² of Eq. (22) becomes an integrated function of goand n as given in Eq. (25). The solid lines in FIG. 12 generated withthe regression (25) agree well with the FE solutions represented bysymbols.

c ² =f _(o) ^(c)(n)f ₂ ^(c)(ε_(o))+f ₁ ^(c)(n)f ₃ ^(c)(ε_(o))1n (100h_(t) /D)   (25)

f _(o) ^(c)(n)=a _(ot) ^(c) n ^(−i) ; a _(ot) ^(c)=(1.09, −1.21, 1.13)

f ₁ ^(c)(n)=a _(1t) ^(c) n ^(−i) ; a _(1t) ^(c)=(0.104, −0.323, 0.405)

f ₂ ^(c)(n)=a _(2t) ^(c)ε_(o) ^(i) ; a _(2t) ^(c)=(1.19, −117, 11500)

f ₃ ^(c)(n)=a _(3t) ^(c)ε_(o) ^(i) ; a _(3t) ^(c)=(0.508, 345, −49500)

[0097]FIG. 13 shows the effect of yield strain on ε_(o) vs. d/D curves.Equivalent plastic strain also decreases with higher yield strain forgiven d/D, since the higher yield strain prolongs the initial dominanceof elastic deformation and delays plastic deformation. Equation (26) isan integrated regression formula extending Eq. (23) to various values ofyield strain. $\begin{matrix}{{ɛ_{p} = {{f_{o}^{ɛ}(n)}{f_{2}^{ɛ}\left( ɛ_{o} \right)}\left( \frac{d}{D} \right)^{{f_{i}^{ɛ}{(n)}}{f_{3}^{ɛ}{(ɛ_{o})}}}}}{{{f_{o}^{ɛ}(n)} = {a_{o\quad i}^{ɛ}n^{- i}}};{a_{o\quad i}^{ɛ} = {{\left( {1.82,{- 5.82},6.92} \right){f_{1}^{ɛ}(n)}} = {a_{1\quad i}^{ɛ}n^{- i}}}};{a_{1\quad i}^{ɛ} = \left( {1.45,{- 0.641},{- 0.233}} \right)}}{{{f_{2}^{ɛ}(n)} = {a_{2\quad i}^{ɛ}ɛ_{o}^{i}}};{a_{2\quad i}^{ɛ} = \left( {1.05,{- 19.2},{- 3850}} \right)}}{{{f_{3}^{ɛ}(n)} = {a_{3\quad i}^{ɛ}ɛ_{o}^{i}}};{a_{3\quad i}^{ɛ} = \left( {0.895,66.7,{- 7090}} \right)}}} & (26)\end{matrix}$

[0098]FIG. 14 shows the effect of yield strain on constraint factor ψvs. d/D curves. FIG. 14(a) and FIG. 14(b) refer to transition and fullyplastic region, respectively. While yield strain strongly affects ψ vs.d/D curve in transition region, it hardly affects the curve in fullyplastic region. For the range of d/D≧0/15, ψ can be given by extendingEq. (24) to varying yield strain as: $\begin{matrix}{{{\psi = {{{f_{o}^{\psi}(n)}{f_{2}^{\psi}\left( ɛ_{o} \right)}} + {{f_{1}^{\psi}(n)}{f_{3}^{\psi}\left( ɛ_{o} \right)}\left( \frac{d}{D} \right)}}},\left( {{d/D} \geq 0.15} \right)}{{{f_{o}^{\psi}(n)} = {a_{o\quad i}^{\psi}n^{- i}}};{a_{o\quad i}^{\psi} = \left( {3.06,{- 4.4},4.19} \right)}}{{{f_{1}^{\psi}(n)} = {a_{1\quad i}^{\psi}n^{- i}}};{a_{1\quad i}^{\psi} = \left( {{- 0.227},0.317,1.25} \right)}}{{{f_{2}^{\psi}(n)} = {a_{2\quad i}^{\psi}ɛ_{o}^{i}}};{a_{2\quad i}^{\psi} = \left( {1.06,{- 30.3},307} \right)}}{{{f_{3}^{\psi}(n)} = {a_{3\quad i}^{\psi}ɛ_{o}^{i}}};{a_{3\quad i}^{\psi} = \left( {3.34,{- 1290},61000} \right)}}} & (27)\end{matrix}$

[0099] Young's modulus in indentation test is primarily determined bythe slope of unloading load-depth curve or by the amount of elasticrecovery. Pharr et al. presumed that unloading load-depth curve isnonlinear, and the initial unloading slope of curve has a close relationwith Young's modulus. Here a determining criterion is set up for initialunloading slope.

[0100]FIG. 15(a) shows load-depth curves obtained by FEA for a varietyof material hardening exponent n with given Young's modulus and yieldstrain. FIG. 15(b) shows corresponding unloading slopes, S. Hereabscissa r=P/Pmax measures the portion of unloading curve used forlinear regression. That is, the portion P_(max) to P is used for linearregression. The slope decreases with increasing regression range r, andit converges to a certain value for r<0.1. Thus this converged value maybe defined as a initial unloading slope, and the slope should bemeasured for r=P/P_(max)<0.1.

[0101] Table 1 compares the slopes obtained from the regression ranger=0.1 and r=0.5 for a variety of material properties. With the slopesfor r=0.1 as references, the slopes for r=0.5 show errors of about 3%.The slope error increases with regression range r. Note again that theslope should be measured with r<0.1, since the slope error amplifies thetotal error of measured Young's modulus in addition to the inherenterror of Young's modulus equations described below. TABLE 1 S S Errorε_(o) E n (r = 0.1) (r = 0.5) (%) 0.002 200  5 9.32 9.11 2.29  7 9.609.37 2.36 10 9.96 9.70 2.69 13 10.1 9.77 3.03  70 10 4.20 4.08 2.86 40010 15.7 15.3 2.31

[0102] Pharr et al. presumed that unloading load-depth curve isessentially nonlinear, and the initial unloading slope, S, of load-depthcurve has a close relation with Young's modulus. They proposed followingYoung's modulus equation. $\begin{matrix}{E = \frac{\left( {1 - \nu^{2}} \right)}{{d/S} - {\left( {1 - \nu_{I}^{2}} \right)/E_{I}}}} & (28)\end{matrix}$

[0103] Here d is the actual projected indentation diameter with materialpile-up/sink-in considered. Eq. (28) was originally derived under theassumption of a rigid cylindrical indenter penetrating the elastic planespecimen. That is, a plane indenter with circular cross-section indentsinto an elastic flat specimen. The concept of effective modulus was thenintroduced to include the deformation of indenter. Effective modulus, ina strict sense, never justifies the deformable indenter. At the instantof unloading, due to preceding plastic deformation, specimen is not flatbut concave with a negative radius of curvature. Thus a correctioncoefficient is introduced into Eq. (28) so that it has the modifiedform: $\begin{matrix}{E = \frac{\left( {1 - \nu^{2}} \right)}{{{d/k_{1}}S} - {\left( {1 - \nu_{I}^{2}} \right)/E_{I}}}} & (29)\end{matrix}$

[0104] Here the correction coefficient k₁ takes the real situation; aspherical deforming indenter, and a non-flat elastic-plastic specimen.To study the implication of k₁, Eq. (29) is recast as: $\begin{matrix}{k_{1} = {\frac{d}{s}\left\{ {\frac{1 - \nu^{2}}{E} + \frac{1 - \nu_{I}^{2}}{E_{I}}} \right\}}} & (30)\end{matrix}$

[0105]FIG. 16(a) shows the variation of k₁ with respect to n for variousvalues of E. It can be observed that k, slightly decreases withincreasing n while k₁ is barely affected by E. For general metals havingn values of 5˜∞, it may be concluded k₁=0.83. FIG. 16(b) reveals that k₁converges to 1 when n approaches to 1. In other words, for an elasticmaterial, k₁ recovers Pharr's suggested value, 1.

[0106]FIG. 17(a) shows FEA relationships between E and S/d for variousvalues of n, when indentation depth is given as h_(t)/D=0.06. FIG. 17(a)suggests that E is an increasing function of S/d, and the coefficientsof the function can be functions of n.

[0107] This observation leads us to: $\begin{matrix}{E = {{a\left( \frac{S}{d} \right)} + {b\left( \frac{S}{d} \right)}^{2}}} & (31)\end{matrix}$

[0108]FIG. 17(b) shows linear variation of b with 1/n, when a is fixedas 6200. Incorporating this linear relation into (31) gives:$\begin{matrix}{E = {{a\left( \frac{S}{d} \right)} + {b\left( \frac{S}{d} \right)}^{2}}} & (32)\end{matrix}$

a=6200, b=(173+8710ε_(o))+(93+21331ε_(o))(1/n)

[0109] Table 2 shows calculated values of Young's modulus from Eq. (5),and corresponding errors. Most of errors are within 2% and maximum erroris 5%. Eq. 5 does not contain Young's modulus and Poisson's ratio ofindenter. However it is never problematic considering fixed Young'smodulus and Poisson's ratio of indenter in experiment. TABLE 2 ε_(o) =0.002 ε_(o) = 0.004 E n S/d E Err (%) S/d E Err (%) 200  5 19.48 2031.66 19.10 207 3.62  7 19.59 202 1.01 19.20 205 2.50 10 19.92 204 2.2319.30 204 1.84 13 19.78 201 0.64 19.58 206 3.16 ∞ 20.09 201 0.74 20.00207 3.61  70 10 8.40  67 −5.00 8.31  67 −4.07 100 10 11.43  97 −2.5311.33  99 −0.78 150 10 15.95 151 0.57 15.59 152 1.03 250 10 23.16 2531.21 22.68 257 2.69 300 10 26.16 302 0.67 25.50 310 3.30 350 10 28.98351 0.31 28.34 357 1.97 400 10 31.46 397 −0.74 30.38 397 −0.82

[0110] Synthesizing the above-mentioned argument, a program is preparedfor evaluating material properties from indentation load-depth curve.The flow chart is shown in FIG. 18. Based on load-depth relationobtained from indentation test, material properties finally produced areYoung's modulus, yield strength and strain hardening exponent.

[0111] In the approach of Haggag et al., each repetition of loading andunloading provides one of stress-strain data points. Thus a singleindentation test usually picks up total only 6-7 data point, whichresults in a rather coarse regression. Haggag's theory requires priormaterial constants from extra tensile tests.

[0112] In the new approach, however, material properties are estimatedusing more than hundreds of data points obtained from a single timeloading followed by unloading. New approach is also free from any extratest. It overall leads us to predict material properties in moreaccurate and simpler manners.

[0113] The load-depth curves are generated by using FE analyses forindentation depth of 6% of indenter diameter. Then, the load-depth curveis fed into the program to evaluate material properties. First, theYoung's modulus E is computed from Eq. (29) by using slope S andinitially guessed values of n and ε_(o). Then, c², ε_(p) and a arecalculated from Eqs. (25-27) as many as the number of load and depthdata. From these, the values of n, K, σ_(o) and ε_(o) are calculatedfrom stress-strain relation. And then updated E, d, c², ε_(p), σ, n, K,σ_(o) and ε_(o) are repeatedly calculated until the updated ε_(o) and nare converged within the tolerance.

[0114] Table 3 compares the predicted with real material properties. Theaverage errors are less than 2% for E and σ_(o), and 3% for n. TABLE 3σ_(o)/E Computed Error Computed Error (×10⁻³) n σ_(o)/E (×10⁻³) (%) n(%) 400/200  5 429/198 7.3/1.0 5.44 8.8  7 414/195 3.4/2.6 7.47 6.7 10404/202 1.1/0.8 10.4 4.0 13 400/200 0.1/0.1 13.1 0.8 200/200 10 192/2134.3/6.6 9.40 6.0 400/400 387/422 3.2/5.5 9.63 3.7 400/100 402/1000.5/0.3 10.2 2.0 800/200 798/202 0.3/0.8 9.92 0.8

[0115]FIGS. 19 and 20 compare predicted and real material curves. Solidline is the material curve used for FEA, and symbol is the predictedstress-strain curve. These comparisons for various values of n and ε_(o)as shown in the figures more than validate our new approach.

[0116] The new theory increases the strain range by a factor of five.Enhancement of related functional equations (25) through (27) and (32),which substantially affect the accuracy of prediction, is in progress.

[0117] A new set of indentation governing equations is proposed based onthe FE solutions. The load-depth curve from indentation testsuccessfully converts to a stress-strain curve. The following remarkscan be drawn from the above investigations.

[0118] (1) A new data acquisition point at 0.4 d apart from indentationcenter is selected. This new point features: i) negligible effect offriction coefficient, ii) quite gentle strain gradient and iii) extendedstrain range by a factor of five.

[0119] (2) The indentation variables c², ε_(p) and ψ were regressed forvarious material properties from FE solutions of indentation analyses.From these, it reveals that the dominant parameters in indentation testare strain hardening exponent n and yield strain ε_(o).

[0120] (3) The previous indentation theories assuming constant c²without regard to indentation depth cannot be applied even for shallowindentation. The pile-up and sink-in parameter c² is generally afunction of indentation depth as well as strain hardening exponent n.

[0121] (4) A new program is developed to evaluate material properties byusing regression formulae of indentation variables c², ε_(p) and ψ. Theload-depth curve is generated by FE analyses for indentation depth of 6%of indenter diameter once for all. The load-depth curves were convertedto stress-strain curves, which provided material properties. The averageerrors of evaluated material properties were 2% for E and σ_(o), and 3%for n.

[0122] (5) The previous theory based on experimental observation anddeformation plasticity theory needs prior material constants from extratensile tests. Moreover, the previous theory derives stress-strainrelation with repetitive loading and unloading. However, our newapproach based on flow theory can predict accurate material propertieswith a single loading followed by unloading without prior materialconstants.

[0123] Based on the theories as discussed above, a non-destructivecompression test utilizing the finite element solutions enables us toperform by the automated indentation system of the present invention

[0124] The automated indentation system is comprised of the three parts:a stepmotor control system (1), measurement instrumentation (2), anddata acquisition system (3) as shown in FIG. 21.

[0125]FIG. 22 illustrates a schematic drawing of automated indentationsystem. Stepmotor (12) is suitable for applying to a micom of thepresent invention due to the pulse digital control, high static torqueand smooth controlling of revolution speed. Therefore, a stepmotor(AS66AC-H50) is used in this system. The stepmotor controller enables tocontrol the acceleration/deceleration and regulate the velocity withrepeatability of 3˜5%. A cylindrical linear actuator consists of a ballscrew (14) (BTK 1404C, THK) and backlash nut (16) to suppress backlash.A flexible coupling (13) (SOH32C) connects the ball screw (14) andstepmotor for constraining the rotation and high repeatability.

[0126] The measurement instrumentation is comprised of a load cell (15),laser displacement sensor (17) for measuring the indentation depth, andball indenter (18).

[0127] The load cells (15) with 200 kgf and 20 kgf of the maximum loadare used and interchangeable. The load cell with 20 kgf is used formeasuring material property of rubber.

[0128] The laser displacement sensor (model Z4M-N30V,OMRON) is used formeasuring the indentation depth. The maximum movement (travelingdistance) of laser displacement sensor (17) is 4 mm and resolution is0.5 μm. The laser displacement sensor (17) is connected parallel to thelinear actuator. The connecting socket lessens the additional complianceof system.

[0129] For a ball indenter (18) as shown in FIG. 23, an integratedspherical indenter made of tungsten carbide (WC) is used for preciselymeasuring the indented depth.

[0130] The Data Acquisition System (3) and Control Box (4) are shown inFIG. 21. The signals from the load cell (15) and laser displacementsensor (17) are amplified and filtered by signal amplifier. Theamplified signal is graphed and stored in a file through PC program. TheData acquisition system (3) and motor controller (1) consist of anotebook PC and a control box (4) integrated for portability ofindentation system.

[0131] The stepmotor (12) can be controlled with PC program as shown inFIG. 24. The moving speed and direction of the stepmotor (12) are alsoable to be adjusted by the control box (4). The control box (4) andprogram displayed on a window in PC enable to control the stepmotor (12)and perform the graphing and storing data of load-depth curve andmaterial properties.

[0132] Dimensions of indentation system are H489×W220×D220 mm. Thediameter of indenter tip is 1 mm. The indentation depth is measured bythe laser displacement sensor which is non-contact optical instrument.For a given indentation depth, the indentation load is determined byindenter tip diameter and specimen material properties. If the obtainedindentation depth from the FEA is a net displacement of indenter tip,the measured depth by the laser displacement sensor includes thecompressed displacement between the indenter tip and the head part ofthe fixed laser displacement sensor. FIG. 25 shows P-h curves from theFEA and tests. The FEA adopted the material properties obtained fromtensile test of the same material. The indentation load is controlled sothat the maximum load of FEA and experiment are identical.Experimentally measured displacement contains additional displacement.Let h_(exp) be the experimental indentation depth, and h_(FEM) be the(actual) depth measured from FEA. Then, the additional displacement dueto system compliance h_(add) is h_(exp)-h_(FEM). FIG. 25 showsexperimentally measured P-h curves after correcting h_(add).Consequently, the result is as same as that of the FEA solutions.

[0133] While the present invention has been described in detail with itspreferred embodiments, it will be understood that its furthermodifications are possible. The present application is thereforeintended to cover any variations, uses or adaptations of the inventionfollowing the general principles thereof, and includes such departuresfrom the present disclosure as come within known or customary practicein the art to which this invention pertains within the limits of theappended claims.

What is claimed is:
 1. An automated indentation system for performing acompression test by loading a compressive indentation load (P), therebycalculating an elastic modulus (E) and a yield strength (σ_(o)), and ahardening exponent (n) from measured indentation depth (h_(t)) andindentation load (P), and unloading slope (S), comprises: a measurementinstrumentation (2) having a load cell (15), laser displacement sensor(17) for measuring the indentation depth, and an integrated ballindenter (18), a data acquisition system (3) having an signal amplifierfor amplifying and filtering signals received from said load cell (15)and laser displacement sensor (17), and a control box (4) beingpre-stored computer programming algorisms for adjusting and controllingmoving speed and direction of said stepmotor (12), performingcalculations and plotting graphs of load-depth curves, strain-stresscurves based on said amplified signal data, and said control box (4) forstoring and retrieving measured signal data, material properties andproduced data.
 2. An automated indentation system as claimed in claim 1,wherein said stepmotor control system (1) further comprises acylindrical linear actuator having a ball screw (14) and backlash nut(16) for suppressing backlash, a flexible coupling (13) being connectedto said ball screw (14) and said stepmotor (12) for constrainingrotation and high repeatability.
 3. An automated indentation system asclaimed in claim 2, wherein said stepmotor control system (1) enables tocontrol acceleration/deceleration of said stepmotor (12) and regulatingvelocity with repeatability of 3˜5%.
 4. An automated indentation systemas claimed in claim 1, wherein said load cell (15) is specified based onthe performance of finite element simulation of indentation test, saidload cells (15) with 200 kgf and 20 kgf of the maximum load are used andinterchangeable.
 5. An automated indentation system as claimed in claim1, wherein said laser displacement sensor for measuring indentationdepth is connected parallel to a linear actuator, and maximum travelingdistance of said laser displacement sensor (17) is 4 mm and resolutionis 0.5 μm.
 6. An automated indentation system as claimed in claim 1,wherein said ball indenter (18) is an integrated spherical indenterbeing made of tungsten carbide (WC) for precisely measuring an indenteddepth, and a diameter of indenter tip is 1 mm.
 7. An automatedindentation system as claimed in claim 1, wherein a measured indentationdepth (h_(exp)) being contained an additional displacement due to systemcompliance (h_(add)) is compensated by a displacement relationshipbetween said measured indentation depth (h_(exp)) and an actualindentation depth (h_(FEM)) obtained from FEA.
 8. A computer programmingalgorism for performing an automated indentation test by loading acompressive indentation load (P) , thereby calculating an elasticmodulus (E) and a yield strength (σ_(o)), and a hardening exponent (n)from measured indentation depth (h_(t)) and indentation load (P), andunloading slope (S), said process of the computer programming algorismcomprises steps of: inputting data of measured indentation depth(h_(t)), load (P) and unloading slope (S) from pre-stored data,computing a Young's modulus (E) from unload slope and initially guessedvalues of n and ε_(o), computing indentation diameters (d) from c²equation as many as the number of load and depth data, computingequivalent plastic strains (ε_(p)) and equivalent stresses (σ) accordingto the calculated indentation diameters (d), computing values of strainhardening exponent (n) and K from stress-strain relation, computing ayield stress (σ_(o)) and strain (ε_(o)), computing updated E, d, c²,ε_(p), σ, n, K, σ_(o), and ε_(o) until the updated ε_(o) and n areconverged within the tolerance, and outputting material properties (E,σ_(o), n) and plotting the stress-strain curve.